Hugo G. Silva
1,*,
Pedro M. Areias
2,§, José E. Garção
2, Nicolas Van Goethem
3, and
Mourad Bezzeghoud
1,21
Geophysics Centre of Évora and
2Physics Department, University of Évora, Portugal;
3MathemaAcs Department, Faculty of Sciences, University of Lisbon, Portugal
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
Initial report of ULF
magnetic emissions
associated with
seismic activity
ULF magneFc field measurements near
(about 7 km) the epicenter of the
imminent Loma Prieta earthquake have
revealed anomalous acFvity almost two
weeks before the earthquake with a
remarkable increase three hours before.
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
12/01/24 4
These experiments possibly indicate a single
mechanism responsible for the EM emissions.
It is based on stress acFvated p-‐type charge
carriers in igneous rocks creaFng a ba]ery
effect that successfully describes such
emissions.
Electric currents streaming out of stressed igneous rocks –
A step towards understanding pre-earthquake
low frequency EM emissions
Friedemann T. Freunda,b,*, Akihiro Takeuchib,c, Bobby W.S. Laub
aNASA Goddard Space Flight Center, Planetary Geodynamics Laboratory, Code 698 Greenbelt, MD 20771, USA bSan Jose State University, Department of Physics, San Jose, CA 95192-0106, USA
cNiigata University, Department of Chemistry, Niigata 950-2181, Japan
Accepted 6 February 2006 Available online 19 May 2006
Abstract
Transient electric currents that flow in the Earth’s crust are necessary to account for many non-seismic pre-earthquake signals, in particular for low frequency electromagnetic (EM) emissions. We show that, when we apply stresses to one end of a block of igneous rocks, two currents flow out of the stressed rock volume. One current is carried by electrons and it flows out through a Cu electrode directly attached to the stressed rock volume. The other current is carried by p-holes, i.e., defect electrons on the oxygen anion sublattice, and it flows out through at least 1 m of unstressed rock to meet the electrons that arrive through the outer electric circuit. The two out-flow currents are part of a battery current. They are coupled via their respective electric fields and fluctuate. Applying the insight gained from these laboratory experiments to the field, where large volume of rocks must be subjected to ever increasing stress, leads us to suggest transient, fluctuating currents of considerable magnitude that would build up in the Earth’s crust prior to major earthquakes. ! 2006 Elsevier Ltd. All rights reserved.
Keywords: Seismo-electromagnetic phenomena; Uniaxial loading; Igneous rock; Positive hole (p-hole); p-type semiconductor; Battery current
1. Introduction
A recent letter in Nature (Gerstenberger et al., 2005) begins with the words: ‘‘Despite a lack of reliable determin-istic earthquake precursors, seismologists have significant predictive information about earthquake activity from an increasingly accurate understanding of the clustering prop-erties of earthquakes.’’ This statement alludes to the fact that, when earthquakes happen, they are chaotic events. Their chaotic character is exacerbated by the heterogeneity of the Earth’s crust, particularly along seismically active plate margins, which are crisscrossed by faults and
pot-marked by deeply buried asperities. When and where a given fault segment will fail depends on processes that take place under kilometers of rock. Unless there is a history of prior seismic activity that leaves a trail of signals, it is unknowable where faults go at depth and when asperities might fail. Therefore, the tools of seismology can cast time and approximate location of the next earthquakes only in terms of rather coarse statistical probability.
However, there is a non-seismological approach, which can add to our knowledge base. In this approach, we attempt to understand the electromagnetic signals, if any, that the earth reportedly sends out before major earthquakes.
The literature is replete with reports of pre-earthquake phenomena. Some of these phenomena can be explained mechanistically by assuming that, when large volumes of rock are squeezed deep underground, they deform
1474-7065/$ - see front matter! 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pce.2006.02.027
*Corresponding author. Address: NASA Goddard Space Flight Center,
Planetary Geodynamics Laboratory, Code 698, Room F321B, Greenbelt, MD 20771, USA. Tel.: +1 301 614 6874; fax: +1 301 614 6522.
E-mail address:[email protected](F.T. Freund).
www.elsevier.com/locate/pce Physics and Chemistry of the Earth 31 (2006) 389–396
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
12/01/24 5
“The first arrival Ames of seismo-‐electric and seismo-‐
magneAc fields clearly indicate that these two fields
are coupled to different propagaAon modes, an
observaAon that is consistent with electrokineAc
theory. Fast longitudinal modes generate only seismo-‐
electric field whereas transverse modes are coupled to
magneAc fields.”
First laboratory measurements of seismo-magnetic conversions in
fluid-filled Fontainebleau sand
C. Bordes,
1L. Jouniaux,
2M. Dietrich,
1J.-P. Pozzi,
3and S. Garambois
1Received 7 September 2005; revised 29 October 2005; accepted 16 November 2005; published 6 January 2006.
[1] Seismic wave propagation in fluid-filled porous
materials induces electromagnetic effects due to small relative pore-fluid motions. In order to detect the seismo-magnetic couplings theoretically predicted by Pride (1994), we have designed a small-scale experiment in a low-noise underground laboratory which presents exceptional electromagnetic shielding conditions. Our experiment included accelerometers, electric dipoles and induction magnetometers to characterize the seismo-electromagnetic propagation phenomena. To assess the electrokinetic origin of the measured electric and magnetic fields, we compared records obtained in dry and fluid-filled sand. Extra care has been taken to ensure the mechanical decoupling between the sand column and the magnetometers to avoid spurious vibrations of the magnetometers and misinterpretations of the recorded signals. Our results show that electric and seismo-magnetic signals are associated with different wave propagation modes, thus emphasizing the electrokinetic
origin of these effects. Citation: Bordes, C., L. Jouniaux,
M. Dietrich, J.-P. Pozzi, and S. Garambois (2006), First laboratory measurements of seismo-magnetic conversions in fluid-filled Fontainebleau sand, Geophys. Res. Lett., 33, L01302, doi:10.1029/2005GL024582.
1.
Introduction
[2] Observations of transient electromagnetic phenomena
accompanying the seismic wave propagation in fluid filled porous media date back at least to the work of Ivanov [1940]. Frenkel [1944] gave the first quantitative explanations of these phenomena in term of electrokinetic effects at the pore scale until Pride [1994] developed a complete theory which prompted further studies.
[3] Early and pioneering work by Martner and Sparks
[1959] and Thompson and Gist [1993] and more recent studies by Takeuchi et al. [1997], Mikhailov et al. [2000], and Garambois and Dietrich [2001] have concentrated on field measurements. Laboratory measurements were notably performed by Zhu et al. [2000], and Zhu and Tokso¨z [2005] whereas numerical simulations were performed by Haartsen and Pride [1997], Garambois and Dietrich [2002] and White [2005].
[4] Two kinds of seismo-electromagnetic effects are to be
distinguished. The dominant contribution we are addressing in this paper corresponds to electrical and magnetic coseismic fields accompanying the body and surface waves. The second kind is generated at physico-chemical properties contrasts and consists of independently propagating electromagnetic waves.
[5] Seismo-electromagnetic studies have generally
concentrated on the measurements of electrical fields as they require only a simple instrumentation. The investiga-tion of seismo-magnetic fields has received much less attention mainly because of the high level of electromagnetic noise affecting the magnetic measurements. In order to minimize these disturbances, we have designed a laboratory experiment within the ultra-shielded chamber of the LSBB low-noise laboratory located in Rustrel, southern France.
[6] This paper describes the experimental apparatus as
well as the first results (seismic, electric and magnetic responses) measured in homogeneous Fontainebleau sand. We show that seismo-magnetic conversions are weak but nevertheless measurable. Moreover, the different apparent velocities characterizing the electric and seismo-magnetic events emphasize that they are associated to different propagation modes.
2.
Experimental Apparatus
[7] The LSBB facilities were originally an underground
launching center for the ground-based component of the French strategic nuclear defense. A characterization of the electromagnetic shielding, performed by Gaffet et al. [2003], using a SQUID magnetometer showed that the noise level is below 2 fT/pffiffiffiffiffiffiHz above 10 Hz.
[8] Our experimental apparatus was located within the
shielded chamber and consisted of a porous sample, a seismic source and sensors. The whole experiment includ-ing the triggerinclud-ing of the mechanical source and the data acquisition was remotely controlled from outside the chamber to suppress electromagnetic perturbations from the instruments. All measurements were performed with a 24 bit seismic recorder (Geometrics StrataVisor NZ) using a
21 ms time sampling rate.
2.1. Seismo-Electromagnetic Measurements
[9] Our experiments were performed with two 1 m high
and 8 cm diameter vertical Plexiglas columns filled with Fontainebleau sand (Figure 1). This sand contains 99% of
silica with grain size smaller than 300 mm. The measured
permeability of the sand is 5.8 10!12m2and its bulk density
1.7727 103 kg/m3, its electrical resistivity is 22 kW.m, and
the water conductivity is 3.1 mS/m with a pH of 6.55 at
GEOPHYSICAL RESEARCH LETTERS, VOL. 33, L01302, doi:10.1029/2005GL024582, 2006
1Laboratoire de Ge´ophysique Interne et Tectonophysique, Universite´
Joseph Fourier, UMR 5559 Grenoble, France.
2Ecole et Observatoire des Sciences de la Terre, Universite´ Louis
Pasteur, UMR 7516 Strasbourg, France.
3Laboratoire de Ge´ologie, Ecole Normale Superieure, Paris, France.
Copyright 2006 by the American Geophysical Union. 0094-8276/06/2005GL024582$05.00
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
12/01/24 6
“In the wet rock experiments, EM emissions in
the HF range were detected accompanied with
AEs, although EMEs in the low-‐frequency range
were not observed. In contrast, in the dry rock
experiments, EMEs in both the high-‐frequency
and low-‐frequency ranges were detected. The
generaAon mechanism can be explained based
on piezoelectricity and compensaAon charges.”
Electromagnetic emissions from dry and wet granite
associated with acoustic emissions
Shingo Yoshida1 and Tsutomu Ogawa
Earthquake Research Institute, University of Tokyo, Tokyo, Japan
Received 17 March 2004; revised 11 June 2004; accepted 25 June 2004; published 11 September 2004.
[1] Triaxial deformation experiments were performed to compare electromagnetic
emissions (EMEs) from dry and wet granite specimens associated with acoustic emissions (AEs). Cylindrical specimens of initially intact granite were tested at room temperature. We measured both high-frequency (10 kHz to 1 MHz) and low-frequency (DC to 500 Hz) components of the electric potential variations. In general, it is difficult to detect EMEs in wet conditions because wet rock has very short electric relaxation time, which is proportional to its electric resistivity. However, in the wet experiment using a granite specimen saturated with distilled water under a constant pore pressure, EMEs in the high-frequency range were detected and accompanied with AEs, although EMEs in the low-frequency range were not observed. In contrast, in the dry granite experiment, EMEs were detected in both the high-frequency and low-frequency ranges. In the high-frequency range, the dry specimen generated EME more frequently than the wet specimen. The generation mechanism can be explained based on piezoelectricity and compensation charges. The difference in the EME generation from wet and dry granite is attributed to the difference of electric relaxation time. Also, electromagnetic fields due to an electric polarization in a dielectric and conductive medium are discussed, including the frequency
dependence of the skin depth. INDEXTERMS: 5109 Physical Properties of Rocks: Magnetic and
electrical properties; 5104 Physical Properties of Rocks: Fracture and flow; 7223 Seismology: Seismic hazard assessment and prediction; KEYWORDS: EME, piezoelectric effect, AE
Citation: Yoshida, S., and T. Ogawa (2004), Electromagnetic emissions from dry and wet granite associated with acoustic emissions, J. Geophys. Res., 109, B09204, doi:10.1029/2004JB003092.
1. Introduction
[2] There have been many articles which report magnetic
and electric fields that might be related to earthquakes. For example, Fraser-Smith et al. [1990] reported that the low-frequency (0.01 Hz) magnetic noise remarkably increased prior to the 1989 Loma Prieta earthquake, measured at Corralitos about 7 km form the epicenter. Nagao et al. [2002] compiled anomalous electromagnetic phenomena at varied frequency ranges, covering ELF to VHS, observed before 1995 Kobe earthquake (M7.2), although it is not certain whether all the observations were truly related to the earthquake, instead of atmospheric activities. The anoma-lous changes were markedly enhances toward the catastro-phe, indicating the possibility that these were caused by the final stage of the earthquake preparation process.
[3] One of plausible mechanisms of generating
electro-magnetic signals associated with earthquakes is the piezo-electric effect. Yoshida et al. [1994, 1997] and Ikeya and Takaki [1996] proposed a model of the generation mecha-nism of the electromagnetic emissions (EMEs) associated
with rock fractures based on the effective polarization of quartz due to an imbalance between the piezoelectric polarization and compensation charges. This model predicts that it is relatively difficult to detect electric signals from wet rock because wet rock has very short electric relaxation time, which is proportional to its electric resistivity. Many laboratory studies have focused on the electric signals produced by the piezoelectric effect of quartz [e.g., Nitsan, 1977; Yoshida et al., 1994, 1997; Ikeya and Takaki, 1996]. These experiments were performed using dry rock speci-mens. It is not known whether or not detectable EMEs are generated from wet rock, while it has been verified that electric signals in wet rock are generated due to electroki-netic effect [e.g., Jouniaux and Pozzi, 1995; Yoshida, 2001]. The present paper aims to compare the EMEs from wet and dry granite associated with acoustic emissions (AEs).
2. Experimental Procedure
[4] We performed triaxial deformation experiments on
initially intact specimens of Inada granite (course grain size) with a quartz content of about 35%, using a triaxial apparatus as described by Yoshida [2001]. During the experiments, we measured differential axial stress, axial piston displacement, AEs, and the high- and low-frequency components of EMEs.
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, B09204, doi:10.1029/2004JB003092, 2004
1
Also at Institute of Physical and Chemical Research, Saitama, Japan. Copyright 2004 by the American Geophysical Union.
0148-0227/04/2004JB003092$09.00
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
We are studying the electrical
properFes of different types of
graniFc samples in order to
perform experiments involving
pressure sAmulated
electromagneAc emissions.
a) Real part of the dielectric constant; b) Imaginary part of the dielectric constant.
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
o
Cauchy equaFon of moFon and Cauchy lemma
o
Mass conservaFon
∂(Jρ)
∂t
= 0
J = det F
∇ · σ = ρ
D ˙u
Dt
n
T
σ = t
with
12/01/24 8MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
With boundary condiFons
n
· [[b]] = 0
n
· [[d]] = 0
n
× [[h − v × d]] = 0
n
× [[e + v × b]] = 0
∇ × e + ˙b = 0
∇ × h + ˙d = 0
∇ · b = 0
∇ · d = 0
Ω
Γ
material
vacuum
o
Maxwell’s equaFons for insulators ρ
free
= 0 and j
free
=
0
.
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
ε =
1
2
(b
− I)
Ogdon model, N=2
(Mooney-‐Rivlin material)
o
Total Helmoltz free energy
Derived to generate the correct expression for the Maxwell stress tensor
ψ(ε, d, h) =
1
2
(1
− D) ε
T:
C : ε +
1
2
µh
T(I + 2ε)h +
1
2�
d
T(I + 2ε)d
−
1
2
�
1
�
d
· d + µh · h
�
tr[ε] + d
· (I : ε)
o
Strain tensor
DeformaFon
energy
ElectromagneFc
terms
Piezoelectric
effect
Where D is the damage variable
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
o
The following loading/unloading condiFons
o
The corresponding damage loading funcFon
ϕ(ε) = (1
− D)ε
1
− ε
max
ϕ(ε)
≤ 0
˙
Dϕ(ε) = 0
˙
D
≥ 0
τ =
∂ψ
∂ε
e =
∂ψ
∂d
b =
∂ψ
∂h
τ =
∂ψ
∂ε
e =
∂ψ
∂d
b =
∂ψ
∂h
τ =
∂ψ
∂ε
e =
∂ψ
∂d
b =
∂ψ
∂h
o
The Kirchhoff stress tensor, the electric field and the magneFc
inducFon
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
o
The first variaFon of τ,
e
and
b
with respect to
ε,
d
and
h
The third derivaAves with respect ε, d and h are also needed for b
δτ =
∂
2
ψ
∂ε
2
: δε +
∂
2
ψ
∂ε∂d
· δd +
∂
2
ψ
∂ε∂h
· δh
δe =
∂
2
ψ
∂d∂ε
: δε +
∂
2
ψ
∂d
2
· δd +
∂
2
ψ
∂d∂h
· δh
δb =
∂
2
ψ
∂h∂ε
: δε +
∂
2
ψ
∂h∂d
· δd +
∂
2
ψ
∂h
2
· δh
12/01/24 12MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
o
IntegraFng in Ω provides the virtual work
The second variaAon of W is also required for the applicaAon of Newton's method.
o
The backward-‐Euler method is used for the integraFon
˙b ∼
=
b
n+1
− b
n
∆t
˙d ∼
=
d
n+1
− d
n
∆t
δW =
�
Ω0τ :
∇δu dΩ
0+
�
Ω0˙b · δh dΩ
0−
�
Ω0(
∇ × δh) · e dΩ
0+
�
Ω0�
∇ × h − ˙d
�
· δd dΩ
0+
r
b�
Ω0∇ · b∇ · δbdΩ
0+ r
d�
Ω0∇ · d∇ · δddΩ
0+
�
Γδλ
Ibn
· [[b]]dΓ +
�
Γδλ
Idn
· [[d]]dΓ+
�
Γδλ
IIdn
× [[h − v × d]]dΓ +
�
Γδλ
IIbn
× [[e + v × b]]dΓ −
�
Γtt
· δudΓ
t+
�
Γe
× δh dΓ = 0
12/01/24 13MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
We use AceGen for the derivaAon of the discreAzed equaAons
o
The 2D discreFzaFon is based on a 3-‐node triangle
o
The δ-‐variaFon of (
Δ
)
δ
∇ (•) = ∇δ (•) − ∇ (•) ∇δu
o
The d-‐variaFon of (
Δ
)
d
∇ (•) = ∇d (•) − ∇ (•) ∇du
Here
is a tensor and
is a spaAal gradient
Δ
u =
�
N
K
u
K
d =
�
N
K
d
K
h
z
=
�
N
K
(h
z
)
K
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
-5.475e+07 1.820e+08 4.188e+08 6.556e+08 8.924e+08 E[X] -4.387e+08 -2.243e+08 -9.914e+06 2.045e+08 4.189e+08 E[Y] -7.626e+07 -3.808e+07 8.790e+04 3.826e+07 7.643e+07 BZ -7.862e+07 -4.230e+07 -5.978e+06 3.034e+07 6.666e+07E[X] -1.788e+08 -8.970e+07 -6.058e+05 8.849e+07 1.776e+08E[Y] -4.278e-02 -2.137e-02 5.234e-05 2.147e-02 4.289e-02BZ 1eï08 40 30 20 10 0 10 1 0.1 0.01 0.001 0.0001 1eï05 1eï06 1eï07 50
0.
4m
˙u = 1 ms
−1
I
23
= 12.71 Cm
−2
I
12
=
−5.207 Cm
−2
I
11
= 15.08 Cm
−2
! = 6
× 10
−9
Fm
−1
µ = 1.256
× 10
−6
Hm
−1
ν = 0.357
E = 69.59 GPa
Imposed displacement
Imposed d
xx1 m
Accumulated number of iterations
∆t = 1× 10−6 s (each step 1% elongation)
∆t = 1× 10−5 s (each step 10% elongation)
∆t = 1× 10−4 s (each step 100% elongation)
Error Original geometry NC−1 NC−1 T -5.475e+07 1.820e+08 4.188e+08 6.556e+08 8.924e+08 E[X] -4.387e+08 -2.243e+08 -9.914e+06 2.045e+08 4.189e+08 E[Y] -7.626e+07 -3.808e+07 8.790e+04 3.826e+07 7.643e+07 BZ -7.862e+07 -4.230e+07 -5.978e+06 3.034e+07 6.666e+07E[X] -1.788e+08 -8.970e+07 -6.058e+05 8.849e+07 1.776e+08E[Y] -4.278e-02 -2.137e-02 5.234e-05 2.147e-02 4.289e-02BZ 1eï08 40 30 20 10 0 10 1 0.1 0.01 0.001 0.0001 1eï05 1eï06 1eï07 50 0. 4m ˙u = 1 ms−1 I23= 12.71 Cm−2 I12=−5.207 Cm−2 I11= 15.08 Cm−2 ! = 6× 10−9Fm−1 µ = 1.256× 10−6Hm−1 ν = 0.357 E = 69.59 GPa
Imposed displacement Imposed dxx
1 m
Accumulated number of iterations ∆t = 1× 10−6s (each step 1% elongation)
∆t = 1× 10−5s (each step 10% elongation) ∆t = 1× 10−4s (each step 100% elongation)
Error Original geometry NC−1 NC−1 T -5.475e+07 1.820e+08 4.188e+08 6.556e+08 8.924e+08 E[X] -4.387e+08 -2.243e+08 -9.914e+06 2.045e+08 4.189e+08 E[Y] -7.626e+07 -3.808e+07 8.790e+04 3.826e+07 7.643e+07 BZ -7.862e+07 -4.230e+07 -5.978e+06 3.034e+07 6.666e+07E[X] -1.788e+08 -8.970e+07 -6.058e+05 8.849e+07 1.776e+08E[Y] -4.278e-02 -2.137e-02 5.234e-05 2.147e-02 4.289e-02BZ 1eï08 40 30 20 10 0 10 1 0.1 0.01 0.001 0.0001 1eï05 1eï06 1eï07 50 0. 4m ˙u = 1 ms−1 I23= 12.71 Cm−2 I12=−5.207 Cm−2 I11= 15.08 Cm−2 ! = 6× 10−9Fm−1 µ = 1.256× 10−6 Hm−1 ν = 0.357 E = 69.59 GPa
Imposed displacement Imposed dxx
1 m
Accumulated number of iterations
∆t = 1× 10−6s (each step 1% elongation)
∆t = 1× 10−5s (each step 10% elongation)
∆t = 1× 10−4s (each step 100% elongation)
Error Original geometry NC−1 NC−1 T -5.475e+07 1.820e+08 4.188e+08 6.556e+08 8.924e+08 E[X] -4.387e+08 -2.243e+08 -9.914e+06 2.045e+08 4.189e+08 E[Y] -7.626e+07 -3.808e+07 8.790e+04 3.826e+07 7.643e+07 BZ -7.862e+07 -4.230e+07 -5.978e+06 3.034e+07 6.666e+07E[X] -1.788e+08 -8.970e+07 -6.058e+05 8.849e+07 1.776e+08E[Y] -4.278e-02 -2.137e-02 5.234e-05 2.147e-02 4.289e-02BZ 1eï08 40 30 20 10 0 10 1 0.1 0.01 0.001 0.0001 1eï05 1eï06 1eï07 50 0. 4m ˙u = 1 ms−1 I23= 12.71 Cm−2 I12=−5.207 Cm−2 I11= 15.08 Cm−2 ! = 6× 10−9Fm−1 µ = 1.256× 10−6Hm−1 ν = 0.357 E = 69.59 GPa
Imposed displacement Imposed dxx
1 m
Accumulated number of iterations ∆t = 1× 10−6s (each step 1% elongation)
∆t = 1× 10−5s (each step 10% elongation) ∆t = 1× 10−4s (each step 100% elongation)
Error Original geometry NC−1 NC−1 T 12/01/24 15
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
Damage:
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
E
x
(V/m):
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
E
y
(V/m):
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
B
z
(T):
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
MathemaFcal modelling and numerical simulaFons for geophysics, 20 June 2011
